Assume r=p/q where p and q are integers with gcd(p,q)=1. Then r^2=6=(P^2)/(q^2), or
Since 2 divides LHS of (1) then it must also divide RHS of (1) i.e. 2 divides p, so p=2k for some integer k. Then (1) gives us
2*3*q*q=2*k*2*k or 3*q*q=2*k*k (2)
But since 2 divides RHS of (2) it must also divide LHS of (1) and therefore it must divide q. But this implies that 2 divides both p and q which is a contradiction to the assumption that gsd(p,q)=1.